HSE Home Hilbert Space Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  HSE Home  >  Th. List  >  speccl Structured version   Unicode version

Theorem speccl 27112
Description: The spectrum of an operator is a set of complex numbers. (Contributed by NM, 11-Apr-2006.) (New usage is discouraged.)
Assertion
Ref Expression
speccl  |-  ( T : ~H --> ~H  ->  (
Lambda `  T )  C_  CC )

Proof of Theorem speccl
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 specval 27111 . 2  |-  ( T : ~H --> ~H  ->  (
Lambda `  T )  =  { x  e.  CC  |  -.  ( T  -op  ( x  .op  (  _I  |`  ~H ) ) ) : ~H -1-1-> ~H }
)
2 ssrab2 3521 . 2  |-  { x  e.  CC  |  -.  ( T  -op  ( x  .op  (  _I  |`  ~H )
) ) : ~H -1-1-> ~H }  C_  CC
31, 2syl6eqss 3489 1  |-  ( T : ~H --> ~H  ->  (
Lambda `  T )  C_  CC )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4   {crab 2755    C_ wss 3411    _I cid 4730    |` cres 4942   -->wf 5519   -1-1->wf1 5520   ` cfv 5523  (class class class)co 6232   CCcc 9438   ~Hchil 26131    .op chot 26151    -op chod 26152   Lambdacspc 26173
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1637  ax-4 1650  ax-5 1723  ax-6 1769  ax-7 1812  ax-8 1842  ax-9 1844  ax-10 1859  ax-11 1864  ax-12 1876  ax-13 2024  ax-ext 2378  ax-sep 4514  ax-nul 4522  ax-pow 4569  ax-pr 4627  ax-un 6528  ax-cnex 9496  ax-hilex 26211
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 974  df-tru 1406  df-ex 1632  df-nf 1636  df-sb 1762  df-eu 2240  df-mo 2241  df-clab 2386  df-cleq 2392  df-clel 2395  df-nfc 2550  df-ne 2598  df-ral 2756  df-rex 2757  df-rab 2760  df-v 3058  df-sbc 3275  df-dif 3414  df-un 3416  df-in 3418  df-ss 3425  df-nul 3736  df-if 3883  df-pw 3954  df-sn 3970  df-pr 3972  df-op 3976  df-uni 4189  df-br 4393  df-opab 4451  df-mpt 4452  df-id 4735  df-xp 4946  df-rel 4947  df-cnv 4948  df-co 4949  df-dm 4950  df-rn 4951  df-iota 5487  df-fun 5525  df-fn 5526  df-f 5527  df-f1 5528  df-fv 5531  df-ov 6235  df-oprab 6236  df-mpt2 6237  df-map 7377  df-spec 27068
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator